Finite Group Databasebeta

← #27281785

Subgroups — #27281785

Order 13568.

448 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283799C211yesmaximal, characteristic
2#20279923C211yesmaximal, characteristic
3#20273433C211yesmaximal, characteristic
4#20273431C211yesmaximal, characteristic
5#20273058C211yesmaximal, characteristic
6#20272571C211yesmaximal, characteristic
7C53×#1285C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10#16810752—22no—
11#16810751—22no—
12#16810680C2211yescharacteristic
13#16810680C2211yescharacteristic
14#16810677—21no—
15#16810673—21no—
16#16810544—21no—
17#16810544—21no—
18#16810529C412yes—
19#16810522—21no—
20#16810522—21no—
21#16810493C412yes—
22#16810370—22no—
23#16810368—22no—
24#16810348—21no—
25#16810348—21no—
26C53×#432C2211yescharacteristic
27C53×#432C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×Dic106C2×C411yescharacteristic
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35C4×Dic106—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38C4×Dic106—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—22no—
41C2×C4⋊Dic53—22no—
42C2×C4⋊Dic53C2×C412yes—
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×#11719919D411yescharacteristic
45C2×#11719919D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C4⋊#88322—21no—
49C4⋊#88322D412yes—
50#13133146—22no—
51#13133146—22no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133128—21no—
55#13133128—21no—
56#13133127—22no—
57C8×Dic53—22no—
58#13133124—22no—
59#13133124—22no—
60C2×C4⋊C4×C53—21no—
61C2×C42×C53C2×C412yes—
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—21no—
64C4⋊C8×C53D412yes—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#97×C53—21no—
68#97×C53—21no—
69#95×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—22no—
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C411yescharacteristic
79C2×Dic106—21no—
80C2×Dic106—21no—
81C2×Dic106—22no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C2×Dic106—41no—
85C2×Dic106—82no—
86C2×C8×C53—22no—
87C2×C8×C53—41no—
88C4⋊C4×C53—24no—
89C42×C53—42no—
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53#4111yescharacteristic
93C42×C53#4111yescharacteristic
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—22no—
97C4⋊Dic53—22no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100C4⋊Dic53—82no—
101C4⋊Dic53—82no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53—21no—
109C4×Dic53—21no—
110C2×#88322—22no—
111C2×#88322—41no—
112C23×C53C2×C22⋊C411yescharacteristic
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—24no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53C4×D411yescharacteristic
121C2×C4×C53C4×D411yescharacteristic
122C2×C4×C53#12011yescharacteristic
123C2×C4×C53#12111yescharacteristic
124C2×C4×C53#12111yescharacteristic
125C2×C4×C53#12311yescharacteristic
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53#10112yes—
139Dic106—42no—
140Dic106—42no—
141Dic106—82no—
142Dic106—82no—
143C8×C53—42no—
144C8×C53—81no—
145#88322—42no—
146#88322—81no—
147#9454—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53#38611yescharacteristic
151C22×C53C2×#10111yescharacteristic
152C22×C53#41311yescharacteristic
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—44no—
159C4×C53#45612yes—
160Dic53—22no—
161Dic53—41no—
162Dic53—41no—
163Dic53—41no—
164Dic53—82no—
165#2983—531no—
166#2567—531no—
167#1556—531no—
168#1556—531no—
169#1467—531no—
170#1285—531no—
171#1285D5311yescharacteristic
172C2×C53—21no—
173C2×C53—21no—
174C2×C53#188611yescharacteristic
175C2×C53#190111yescharacteristic
176C2×C53#191011yescharacteristic
177C2×C4⋊Q8—531no—
178C2×C4×Q8—531no—
179#446—1062no—
180#446—1062no—
181#432—531no—
182#432—531no—
183#432—1061no—
184#432C2×D5311yescharacteristic
185#432C2×D5311yescharacteristic
186#431—1061no—
187#395—1061no—
188#395—1061no—
189#391—532no—
190#389—1061no—
191#389—1061no—
192#384—532no—
193C4×C4⋊C4C2×D5311yescharacteristic
194#332—1062no—
195#332—1062no—
196#324—1061no—
197#324—1061no—
198C53#945411yescharacteristic, complemented, Sylow
199C22×Q8—531no—
200C22×Q8—1061no—
201C4⋊Q8—1061no—
202C4⋊Q8—1061no—
203C4⋊Q8—1061no—
204C4⋊Q8—1061no—
205C4×Q8—1061no—
206C4×Q8—1061no—
207C4×Q8—1061no—
208C4×Q8—1061no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—532no—
214C2×C4⋊C4—1062no—
215C2×C4⋊C4—1062no—
216C2×C4⋊C4—1061no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42C4×D5312yes—
220C2×C42C22×D5311yescharacteristic
221C4⋊C8—21no—
222C4⋊C8—532no—
223C4⋊C8—1062no—
224C4⋊C8—1062no—
225C4⋊C8—1061no—
226C4⋊C8D21212yes—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—1061no—
230C22⋊C8—1061no—
231#97—21no—
232#97—21no—
233#97—1062no—
234#97—1061no—
235#97—1061no—
236C4×C8—1062no—
237#95—21no—
238#95—1062no—
239#95—1062no—
240C2×Q8—1062no—
241C2×Q8—1061no—
242C2×Q8—1061no—
243C2×Q8—4242no—
244C2×Q8—2121no—
245C2×Q8—2121no—
246C2×Q8—2121no—
247C22×C4—22no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—1062no—
253C22×C4C2×C4×D5311yescharacteristic
254C22×C4D4×D5311yescharacteristic
255C22×C4D4×D5311yescharacteristic
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—1062no—
259C2×C8—2121no—
260C4⋊C4—24no—
261C4⋊C4—1061no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1062no—
268C4⋊C4—1062no—
269C4⋊C4—4242no—
270C4⋊C4—4242no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C42—42no—
274C42—1061no—
275C42—1061no—
276C42—1061no—
277C42—1061no—
278C42C2×D21211yescharacteristic
279C42#1171994511yescharacteristic
280C42D4×D5311yescharacteristic
281C42#1171994911yescharacteristic
282C23C22⋊C4×D5311yescharacteristic
283Q8—4242no—
284Q8—4242no—
285Q8—2122no—
286Q8—2122no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—532no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—2122no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4#1313317911yescharacteristic
308C2×C4C4×D21211yescharacteristic
309C2×C4#1313318711yescharacteristic
310C2×C4#1313319011yescharacteristic
311C2×C4#1313319211yescharacteristic
312C2×C4#1313320211yescharacteristic
313C8—42no—
314C8—81no—
315C8—4241no—
316C8—2122no—
317C22—21no—
318C22—21no—
319C22#1681050111yescharacteristic
320C22#101×D5311yescharacteristic
321C22#1681060411yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—44no—
328C4—1062no—
329C4—4242no—
330C4—2121no—
331C4—2121no—
332C4—2121no—
333C4#1681080712yes—
334C2—21no—
335C2—21no—
336C2#2027510711yescharacteristic
337C2#2027518211yescharacteristic
338C2#2027524111yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1285C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1285D53
hypocenterC53#9454
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
198. C53#9454semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202837992
#202799232
#202734332
#202734312
#202730582
#202725712
C53×#12852
#945453